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Chapter 18: Problem 41

For diatomic carbon dioxide gas \(\left(\mathrm{CO}_{2},\right.\) molar mass \(\left.44.0 \mathrm{~g} / \mathrm{mol}\right)\) at \(T=300 \mathrm{~K},\) calculate (a) the most probable speed \(v_{\mathrm{mp}} ;\) (b) the average speed \(v_{\mathrm{av}} ;\) (c) the root-mean-square speed \(v_{\mathrm{rms}}\).

Short Answer

Expert verified
The most probable speed is approximately 388.71 m/s. The average speed is approximately 412.04 m/s. The root-mean-square speed is approximately 427.06 m/s.

Step by step solution

01

Convert the molar mass to kg/mol

The value given is in grams, so it must be converted to kilograms: \(44.0 \mathrm{~g/mol} = 0.044 \mathrm{~kg/mol}\).
02

Calculate the most probable speed

Insert the values into the most probable speed formula: \(v_{\mathrm{mp}} = \sqrt{(2*8.314*300)/0.044} = 388.71 \, m/s\).
03

Calculate the average speed

For the average speed, use the given formula: \(v_{\mathrm{av}} = \sqrt{(8*8.314*300)/(3.142*0.044)} = 412.04 \, m/s\).
04

Calculate the root mean square speed

Similar to steps 2 and 3, use the given values in the formula for the root-mean-square speed: \(v_{\mathrm{rms}} = \sqrt{(3*8.314*300)/0.044} = 427.06 \, m/s\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Molecular Speed
Understanding molecular speed is crucial in the kinetic theory of gases, which describes how gas molecules move and interact. Molecules in a gas are in constant random motion, colliding with each other and the walls of their container. Different molecules in a sample at a given temperature will have different speeds due to their random motion.

Molecular speed refers to how fast these molecules are moving. It can be influenced by the temperature and the mass of the molecules. As temperature increases, the average speed of the molecules also increases. The heavier the molecules, the slower they move on average at a given temperature.
Most Probable Speed
Most Probable Speed, often denoted as \(v_{\text{mp}}\), is the speed that the largest number of molecules in a gas sample have at a particular temperature. It is derived from the Maxwell-Boltzmann distribution, which describes the distribution of speeds for molecules in a gas.

The formula for most probable speed is:
  • \( v_{\text{mp}} = \sqrt{\frac{2RT}{M}} \)
Where:
  • \( R \) is the ideal gas constant (8.314 J/mol路K)
  • \( T \) is the temperature in Kelvin
  • \( M \) is the molar mass in kg/mol
At 300 K for \(CO_2\), the most probable speed is approximately 388.71 m/s.
Average Speed
The average speed, denoted by \(v_{\text{av}}\), is calculated by taking the average of all the individual speeds of molecules in a gas sample. It gives a sense of the central tendency of molecular speeds at a given temperature.

The formula for average speed is:
  • \( v_{\text{av}} = \sqrt{\frac{8RT}{\pi M}} \)
Where:
  • \( R \) is the ideal gas constant (8.314 J/mol路K)
  • \( T \) is the temperature in Kelvin
  • \( M \) is the molar mass in kg/mol
At 300 K for \(CO_2\), the average speed calculated is about 412.04 m/s. This speed is greater than the most probable speed, as considering the distribution of speeds, faster speeds have more weight when averaging.
Root-Mean-Square Speed
The root-mean-square (rms) speed, represented by \(v_{\text{rms}}\), measures the square root of the average of the squares of individual molecular speeds. It is often used in calculations involving kinetic energy because it directly relates to the energy of the molecules.

The formula used is:
  • \( v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \)
Where:
  • \( R \) is the ideal gas constant (8.314 J/mol路K)
  • \( T \) is the temperature in Kelvin
  • \( M \) is the molar mass in kg/mol
For \(CO_2\) at 300 K, we compute the rms speed as around 427.06 m/s. Note that rms speed is the largest among most probable and average speeds because it emphasizes higher speeds more due to squaring.
Diatomic Molecules
Even though \(CO_2\) is not a diatomic molecule, understanding diatomic molecules is key in kinetic theory. Diatomic molecules consist of two atoms. Common examples include hydrogen \(H_2\), nitrogen \(N_2\), and oxygen \(O_2\).

Such molecules have rotational and vibrational degrees of freedom which can affect their behavior in gases. These additional degrees of freedom alter the specific heat capacities of gases and their energy distributions. For ideal gases, at moderate temperatures, these effects become more relevant.
Gas Laws
The behavior of gases is well-described by gas laws, which relate various properties such as pressure, volume, and temperature. These laws include:
  • Boyle's Law: At constant temperature, the pressure of a gas is inversely proportional to its volume.
  • Charles's Law: At constant pressure, the volume of a gas is directly proportional to its temperature.
  • Avogadro's Law: At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles of gas.
  • Ideal Gas Law: An equation of state for ideal gases given by \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the moles of gas, \(R\) is the ideal gas constant, and \(T\) is temperature.
These laws are crucial for predicting gas behavior under different conditions and are foundational for understanding the kinetic theory of gases.

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Most popular questions from this chapter

During a test dive in \(1939,\) prior to being accepted by the U.S. Navy, the submarine Squalus sank at a point where the depth of water was \(73.0 \mathrm{~m}\). The temperature was \(27.0^{\circ} \mathrm{C}\) at the surface and \(7.0^{\circ} \mathrm{C}\) at the bottom. The density of seawater is \(1030 \mathrm{~kg} / \mathrm{m}^{3}\). (a) A diving bell was used to rescue 33 trapped crewmen from the Squalus. The diving bell was in the form of a circular cylinder \(2.30 \mathrm{~m}\) high, open at the bottom and closed at the top. When the diving bell was lowered to the bottom of the sea, to what height did water rise within the diving bell? (Hint: Ignore the relatively small variation in water pressure between the bottom of the bell and the surface of the water within the bell.) (b) At what gauge pressure must compressed air have been supplied to the bell while on the bottom to expel all the water from it?

\(\mathrm{A}\) welder using a tank of volume \(0.0750 \mathrm{~m}^{3}\) fills it with oxygen (molar mass \(32.0 \mathrm{~g} / \mathrm{mol}\) ) at a gauge pressure of \(3.00 \times 10^{5} \mathrm{~Pa}\) and temperature of \(37.0^{\circ} \mathrm{C}\). The tank has a small leak, and in time some of the oxygen leaks out. On a day when the temperature is \(22.0^{\circ} \mathrm{C},\) the gauge pressure of the oxygen in the tank is \(1.80 \times 10^{5} \mathrm{~Pa}\). Find (a) the initial mass of oxygen and (b) the mass of oxygen that has leaked out.

A vertical cylinder of radius \(r\) contains an ideal gas and is fitted with a piston of mass \(m\) that is free to move (Fig. \(\mathbf{P 1 8 . 7 7}\) ). The piston and the walls of the cylinder are frictionless, and the entire cylinder is placed in a constant-temperature bath. The outside air pressure is \(p_{0}\). In equilibrium, the piston sits at a height \(h\) above the bottom of the cylinder. (a) Find the absolute pressure of the gas trapped below the piston when in equilibrium. (b) The piston is pulled up by a small distance and released. Find the net force acting on the piston when its base is a distance \(h+y\) above the bottom of the cylinder, where \(y \ll h\). (c) After the piston is displaced from equilibrium and released, it oscillates up and down. Find the frequency of these small oscillations. If the displacement is not small, are the oscillations simple harmonic? How can you tell?

Modern vacuum pumps make it easy to attain pressures of the order of \(10^{-13}\) atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. (a) At a pressure of \(9.00 \times 10^{-14}\) atm and an ordinary temperature of \(300.0 \mathrm{~K}\), how many molecules are present in a volume of \(1.00 \mathrm{~cm}^{3} ?\) (b) How many molecules would be present at the same temperature but at 1.00 atm instead?

You blow up a spherical balloon to a diameter of \(50.0 \mathrm{~cm}\) until the absolute pressure inside is 1.25 atm and the temperature is \(22.0^{\circ} \mathrm{C}\). Assume that all the gas is \(\mathrm{N}_{2},\) of molar mass \(28.0 \mathrm{~g} / \mathrm{mol}\). (a) Find the mass of a single \(\mathrm{N}_{2}\), molecule. (b) How much translational kinetic energy does an average \(\mathrm{N}_{2}\) molecule have? (c) How many \(\mathrm{N}_{2}\) molecules are in this balloon? (d) What is the total translational kinetic energy of all the molecules in the balloon?
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